t. aste, d. boose and n. rivier- from one cell to the whole froth: a whole map

8/3/2019 T. Aste, D. Boose and N. Rivier- From One Cell to the Whole Froth: A Whole Map

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arXiv:cond-mat/9612084v1

[cond-ma

t.dis-nn]9Dec1996

FROM ONE CELL TO THE WHOLE FROTH: A DYNAMICAL MAP

T. Aste , D. Boose and N. Rivier

Laboratoire de Physique Theorique, Universite Louis Pasteur67084 Strasbourg, France

ABSTRACT

We investigate two and three-dimensional shell-structured-inflatable froths, whichcan be constructed by a recursion procedure adding successive layers of cells around agerm cell. We prove that any froth can be reduced into a system of concentric shells.There is only a restricted set of local configurations for which the recursive inflationtransformation is not applicable. These configurations are inclusions between succes-

sive layers and can be treated as vertices and edges decorations of a shell-structure-inflatable skeleton. The recursion procedure is described by a logistic map, whichprovides a natural classification into Euclidean, hyperbolic and elliptic froths. Frothstiling manifolds with different curvature can be classified simply by distinguishing be-tween those with a bounded or unbounded number of elements per shell, without anya-priori knowledge on their curvature. A new result, associated with maximal ori-entational entropy, is obtained on topological properties of natural cellular systems.The topological characteristics of all experimentally known tetrahedrally close- packedstructures are retrieved.

I. INTRODUCTION

A froth is a (topologically stable) division of space by cells, which are convexpolytopes (polygons in 2D, polyhedra in 3D) of various shapes and sizes. These geo-metrical systems have attracted much attention in recent years, both theoretically andexperimentally [1], [2]. The aim in this work is to study a specific class of froths,namely those which are reducible to a set of concentric shells. These particular frothsare structured as if constructed in the following way. In a first stage, cells are addedto a germ cell, forming around it a first layer whose external surface constitutes thesecond shell. In a second stage, cells are added to the first shell so as to form a secondlayer of cells encircling the first one, and so on. We emphasize that the words germ

and stage are purely pictorial and do not imply any particular mode of growth sinceany cell of a generic shell-structured froth may play the role of its germ cell. Such afroth is called shell-structured-inflatable from now on.

On leave from C.I.I.M., Universita di Genova, Genova, Italy

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A definition of a shell-structured-inflatable froth requires the notion of topologicaldistance between cells. The topological distance t between two cells A and B is definedas the smallest number of edges crossed by a path connecting A and B. The germ cellis therefore at distance t = 0. A shell (t) is defined as the interface between two sets ofcells distant by t and t + 1 from the germ cell. A 2D froth is shell-structured-inflatable

if it satisfies the following two conditions :1) For any set cells, equidistant to the germ cell, there exists a closed non self- inter-secting path which goes only through these cells and connects all of them.2) Any cell at distance t from the germ cell is the neighbour of at least one cell atdistance t + 1.Two consecutive shells (t) and (t+1) of a shell-structured-inflatable froth are connectedthrough a set of disjoint edges with one vertex on shell (t) and the other on shell (t+1).These two shells are closed loops of edges delimiting the layer (t + 1) of cells which areat distance t + 1 from the germ cell. Shell (t) divides the froth into an internal froth,constituted of cells at distances r t, and an external froth, with cells at distances

r > t. The extension to a 3D shell-structured-inflatable froths is straightforward andis given in Appendix B.2.In this paper we prove that the 2D and 3D shell-structured-inflatable froths are

constructed according to a recursion procedure which is the logistic map [3], well-known in the theory of dynamical systems. The logistic map provides a natural clas-sification of these froths according to the behaviour of the number of edges per shellas the topological distance t increases.

Any given froth is not necessarily shell-structured-inflatable. However, it has to benoted that a froth can always be decomposed into shells with respect to an arbitrarilychosen germ cell. In this decomposition, each cell of the layer (t) belongs to one of twocategories. The cells of the first category, individually, have neighbours in both layers

(t 1) and (t + 1) and, collectively, are building up a complete ring around the chosengerm cell. The set of all these rings constitutes the skeleton of the shell-structure.The cells of the second category have neighbours in only one of the two layers (t 1)or (t + 1). These cells can be considered as local topological defects included betweenthe rings of the skeleton of the shell-structure. The skeleton is itself a space-fillingfroth which is shell-structured-inflatable. The recursion procedure that we are studyingapplies to such a structure.

The plan of this paper is the following. In Section 2, we derive the recursion pro-cedure associated with 2D shell-structured- inflatable froths and show that it can bewritten as the logistic map. The resulting classification into Euclidean, hyperbolic and

elliptic froths is discussed. In Section 3, it is shown that the recursion procedure in 3Dis again described by a logistic map. The curvature of the embedding space is classi-fied as for the 2D froths. Section 4 gives examples of space-filling cellular structureswhich fit into the classification of 3D shell-structured-inflatable froths provided by thelogistic map. In Section 5, a new bound on topological properties of natural cellular

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structures is obtained. The topological properties of all experimentally known tetra-hedrally close-packed (t.c.p.) structures are retrieved under the hypothesis of shell-reducibility. A conclusion emphasizes the main results of the paper. In Appendix A,the recursion procedure is generalized to 2D shell-structured- inflatable networks withvertex coordination larger than 3. Local topological defects in 2 and 3D shell-reducible

but not inflatable froths are considered in Appendix B. Random 3D Euclidean frothsare constructed from 2D random shell networks in Appendix C.

II. RECURSION PROCEDURE FOR 2D FROTHS

This section is concerned with two dimensional shell-structured-inflatable froths.The recursion procedure is derived here for froths and it is extended to networks withvertex coordination larger than 3 in Appendix A. Fig.(1) shows an example of a frothwith the various shells indicated by bold lines and labelled by the index t (the shell

t = 0 corresponding to the boundary of the germ cell). Let V(t)+() be the number

of vertices going out from shell (t) towards shell (t + 1) (towards shell (t

1)). LetF(t) be the number of cells in the layer between shells (t) and (t + 1). Ifn is theaverage number of edges per cell in layer (t), the edges in this layer are accounted for,as follows

nF(t) = V(t)

+ 2V(t)+ + 2V

(t+1)

+ V(t+1)+ . (2.1)

In the right-hand-side of this equation, the quantity V(t)

+ V(t)+ is the total number

of vertices constituting shell (t), the quantity V(t+1)

+ V(t+1)+ is the total number of

vertices constituting shell (t + 1) whereas the quantity V(t+1)

+ V(t)+ gives the number

of vertices (counted twice) bounding the edges separating the cells comprised between

shells (t) and (t + 1). Since V(t)+ = V

(t+1)

and F(t) = V(t)+ , one has the recursion

equationnV(t)+ = 4V(t)+ + V(t+1)+ + V(t) . (2.2)

The matrix form of this recursion equation is

V(t+1)+

V(t+1)

=

s 11 0

V(t)+

V(t)

, (2.3)

with recursion parameter s = n 4. Eq.(2.3) generates recursively the whole frothfrom the germ cell. In general, the quantity n changes from one layer to the next.Hence the recursion parameter should depend on the distance t. However, the valueof n associated to a layer of cells at a distance t from the germ cell must, as t ,converge to the average value for any cell in the froth. Moreover, since the choice ofthe germ cell is completely arbitrary, the quantity n associated with layer (t) is anaverage. Consequently, the recursion parameter can be taken as an effective quantity

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which is independent oft, and the quantity n is then the average number of edges percell in the froth. The initial conditions in Eq.(2.3) are then V

(0)

= 0 and V(0)+ = n.

The recursion procedure described by Eq.(2.3) appears also in other instances,such as in the computation by decimation of the electronic energy spectrum in the1D tight- binding model [4]. In this case, the variables V(t) are replaced by the

components of the electronic wave-functions in the basis of the site states, and therecursion parameter s is the (dimensionless) energy of the electron.

Eq.(2.3) gives an immediate link between the shell-structured-inflatable froths

and the logistic map. Indeed, from the relations sV(t)+ = V

(t+1)+ + V

(t1)+ , sV

(t+1)+ =

V(t+2)+ + V

(t)+ , and sV

(t1)+ = V

(t)+ + V

(t2)+ , one gets

s1V(t)+ = V

(t+2)+ + V

(t2)+ , (2.4)

withs1 = s

2 2 , (2.5)and a similar relation for the Vs. Iterating j times, one obtains

sjV(t)+ = V

(t+2j)+ + V

(t2j)+ , (2.6)

withsj+1 = s

2j 2 , (2.7)

and s0 = s. Eq.(2.7) is the trace map of the recursion matrix in Eq.(2.3). It is alogistic map [3], with two (unstable) fixed points s = 2 and s = 1. The logistic mapdecomposes the axis of values of the recursion parameter s into two different regions.Any point in the region

|s

|> 2 is sent towards infinity by the successive iterations

of the logistic map. By contrast, if |s| < 2, successive iterations of the logistic mapremain all within this interval. The existence of these two intervals classifies all 2Dshell-structured-inflatable froths. This classification, corresponds to the curvature ofthe manifold which the froth tiles. The space is elliptic for |s| < 2, hyperbolic for|s| > 2 and Euclidean for the fixed point s = s = 2. The map relate successivenumbers V(t) of vertices per shell. Iterations of Eq.(2.3) generate trajectories in the

plane (t, V+) starting from the initial points V(1)+ = V

(0)

= 0 and V(0)+ = n.

When |s| < 2, the trajectories are given by the equation

V(t)+ = V

(0)+

sin((t + 1))

sin , (2.8)

with cos() = s/2. Eq.(2.8) shows that all trajectories are finite and end at the point

V(T)+ = 0, with T =

1. Moreover, the values of V(t)+ are bounded by the quantityV(0)+ /sin(). These finite and bounded trajectories are describing the iterative tiling of

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compact manifolds with positive curvature. Indeed, consider a froth tiling the surfaceof a sphere. Suppose that the north pole of the sphere is located in the germ cell;the successive shells are the parallels on the sphere. The number of vertices per shellincreases between the north pole and the equator, then decreases from the equatorto the south pole where the tiling ends. This is precisely the behaviour described by

Eq.(2.8). The quantity T+ 1 = is the topological distance between both poles. Here

are a few examples of regular froths with |s| < 2. To s = 1 corresponds to a frothmade with four triangles, i.e. the surface of a tetrahedron. The recursion parameters = 0 corresponds to a froth made with six squares, i.e. the surface of a cube. Finally,s = 1 is associated with a froth which is the surface of a dodecahedron.

In the case |s| > 2, the solution of Eq.(2.3) is

V(t)+ = V

(0)+

sinh((t + 1))

sinh , (2.9)

with cosh() = s/2. Eq.(2.9) shows that, contrary to the previous case, the values of

V(t)+ increases exponentially with t. All trajectories are now infinite and unboundedin the plane (t,V+). They are therefore describing the iterative tiling of non-compactmanifolds with negative curvature.

At the fixed point s = s = 2, Eq.(2.3) has the solution

V(t)+ = (t + 1)V

(0)+ , (2.10)

The values of V(t)+ have again no upper bound, but here they are increasing linearly

with t as expected for the Euclidean plane by simple geometrical considerations. Thefixed point s = 2 describes shell-structured-inflatable froths covering the Euclidean

plane with cells with 6 edges on average. An example of such froths is the hexagonaltiling.We have shown that the logistic map provides, in a natural way, the topological

classification of tilings of manifolds without any a-priori knowledge of their Gaussiancurvature. In 2D this classification by the logistic map is identical to that provided bythe combination of the Gauss-Bonnet theorem [5] and Eulers equation:

da =

3(6 n)F =

3(2 s)F , (2.11)

here, is the Gaussian curvature and it is integrated over the whole manifold. F is the

total number of cells in the manifold. The tiled manifold is hyperbolic, Euclidean orelliptic according when the integrated curvature is negative, zero or positive, i.e. whenthe recursion parameter s is larger, equal to or smaller than 2. However, the logisticmap is also applicable in 3D where there is no Gauss-Bonnet theorem and the Eulerequation is homogeneous [6].

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III. RECURSION PROCEDURE FOR 3D FROTHS

This Section extends the analysis of the previous one to 3D shell-structured- in-flatable froths. The froth has V vertices, E edges, F faces and C polyhedras. Everyshell of the 3D froth is built up from two superposed different two- dimensional froths,

and looks like a corrugated sphere. This is the same as in 2D, where a shell can beregarded as the superposition of two 1D froths, one whose vertices are connected tothe incoming edges from shell (t 1) to shell (t), and the other, whose vertices areconnected to the outgoing vertices pointing from shell (t) towards shell (t +1). Simi-larly, every spherical shell (t) of the 3D froth is built up of the superposition of two 2Dfroths, one whose edges are connected to the incoming faces of layer (t 1), and theother whose edges are connected to the outgoing faces of layer (t). Let V

(t)+() and

E(t)+() be the numbers of vertices and edges of shell (t), bounding the cells layer (t)

between shells (t) and (t +1) (layer (t 1) between shells (t) and (t 1), respectively),which are making the outgoing (incoming) froth. Let F

(t)+() be the number of

faces of such froths. Both froths are characterized by the identities

V(t)+() E

(t)+() + F

(t)+() = 2 , (3.1)

(Eulers formula) and

3V(t)+() = 2E

(t)+() (3.2)

(since in both 2D froths, any vertex is connected by three edges and any edge isbounded by two vertices).

One has the following relations between two successive shells

V(t+1) = V

(t)+

E(t+1)

= E(t)+

F(t+1)

= F(t)+ .

(3.3)

Shell (t) is a spherical surface tiled by a network with F(t)N faces. One has

F(t+1)N = fF(t)+ 2E(t)+ F(t)N . (3.4)

In this equation, f is the average number of faces per cell in the layer (t).Since the whole shell (t) is a polyhedron, both Eulers formula and the incidence

relations are applicable between the number of edges E(t)N , the number of vertices V(t)N

and the number of faces F(t)N of the shell- polyhedron, namely,

V(t)N E(t)N + F(t)N = 2 (3.5)

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andnNF(t)N = 2E(t)N . (3.6)

Here, nN is the average number of edges per face of the shell- polyhedron. Since itis an elliptic tiling with vertex coordination 3, then nN < 6. The shell-network isthe superposition of two 2D froths, it has therefore 3-connected vertices V+() corre-sponding to the outgoing (incoming) froth, and also 4- connected vertices V atthe intersections between edges of the two 2D froths. The three types of vertices arerepresented in Fig.(2). Fig.(3) shows the shell-network in the particular case of theKelvin froth [7], [8], and indicates a 3-connected vertex and a 4-connected vertex.

The total number of vertices V(t)N on shell (t) is the sum of all 3- and 4- connected

vertices, i.e.

V(t)N = V

(t)+ + V

(t)

+ V(t)

. (3.7)

The total number of edges E(t)N on shell (t) satisfies the equation

2E(t)

N= 3V

(t)

++ 3V

(t)

+ 4V(t)

. (3.8)

Using Equs.(3.5), (3.6) and (3.8), one obtains

V(t)N = 2 +

1

2

1 2nN

(3V

(t)+ + 3V

(t)

+ 4V(t)

) . (3.9)

Combining Equs.(3.7) and (3.9), it is possible to express the variable V(t)

in terms of

the variables V(t)+ and V

(t)

alone as

2V(t)

=4nN

4

nN

(V(t)+ + V(t) )6 nN4

nN , (3.10)

which, with Eq.(3.6) and (3.8), yields

F(t)N =

8 (V(t)+ + V(t) )4 nN . (3.11)

Putting Eq.(3.11) into Eq.(3.2), we obtain, with the help of Eqs.(3.1), (3.3) and (3.4),the following relation

V(t+1)+ =

1

2(f 6)(nN 4) 4

V(t)+ V(t) + 2

8 + f(nN 4)

. (3.12)

Finally, by shifting the variables V+() as

V(t)+() = V

(t)+()

4

8 + f(nN 4)8 (f 6)(nN 4)

, (3.13)

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one obtains the recursion equation

sV(t)+ = V

(t+1)+ + V

(t1)+ , (3.14)

with the recursion parameter

s =1

2

(f 6)(nN 4) 4

. (3.15)

This recursion equation has the same matrix form as in the 2D case

V(t+1)+

V(t+1)

=

s 11 0

V(t)+

V(t)

, (3.16)

As in 2D f and nN can be supposed to be independent of the distance t. Thevariation of the 3D recursion parameter s (the trace of the transfer matrix Eq.(3.16) )is described by the logistic map (2.7), as in the 2D case. Consequently, the classificationof 3D shell-structured-inflatable-froths is the same as in 2D.

Elliptic shell-structured-inflatable froths are associated with |s| < 2. They aretiling iteratively 3D compact manifolds with positive curvature. Indeed, the corre-sponding solution of Eq.(3.16) is finite and bounded in the (t, V) plane

V(t)+ = A sin(t + B) + 2

8 + f(nN 4)2 s

, (3.17)

with cos() = s/2. The coefficients A and B can be deduced from the initial conditions

V(0)+ = 2(f 2) and V

(1)+ = 0.

Hyperbolic shell-structured-inflatable froths are associated with |s| > 2. Theyare tiling iteratively 3D non-compact manifolds with negative curvature. Indeed, thecorresponding solution of Eq.(3.16) is unbounded in the (t, V) plane

V(t)+ = A sinh(t + B) + 2

8 + f(nN 4)2 s

, (3.18)

with cosh() = s/2. As previously, the coefficients A and B can be determined fromthe initial conditions.

For s =

2 the solution reads

V(t)+ = (1)t

At + B

+

8 + f(nN 4)2

. (3.19)

with A and B deducible from the initial conditions.

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The solution of Eq.(3.16) associated to the fixed point s = s = 2 is

V(t)+ = (t + 1)

V(0)+ + t

8 + f(nN 4)

. (3.20)

The quadratic dependence in t is the one expected from simple geometrical reasoningfor a tiling of the 3D Euclidean space.

As in 2D, the logistic map gives a natural description of tilings of three dimen-sional manifolds without the need of any a-priori information on their curvature. Con-sequently, the logistic map is able to characterize curved manifolds even when theGaussBonnet formula is not applicable [9], [10]. The generation of tilings of curvedmanifold by the recursion procedure has therefore a wider applicability than the GaussBonnet formula.

IV. EXAMPLES OF 3D SHELL-STRUCTURED

INFLATABLE FROTHS

In order to illustrate the previous considerations, we give some known examples of3D froths and show that they fit our classification. All are monotiled (i.e. constitutedof topologically identical cells), apart from the last example.

The only regular elliptic froths in 3D are {3, 3, 3} (packing of tetrahedra), {4, 3, 3}(packing of cubes) and {5, 3, 3} (packing of dodecahedra) [11]. They correspond tos = 1, s = 2 and s = 1 respectively. Note that the case s = 0 does not correspond toany regular froth. Indeed, the only solution s = 0 of Eq.(3.15) with f and nN < 6being both integers is f = 10, nN = 5; which is not regular.

Consider Eq.(3.15) in the Euclidean case (i.e. s = 2). This equation gives arelationship between the average number of neighbours per cell (f) in the 3D frothand the average number of edges per cell (nN) in the 2D spherical shell-network

f = 6 + 8nN 4 . (4.1)

This equation gives the condition for Euclidean space-filling by a shell-structured-inflatable froth. Note that, from Eq.(4.1), the minimal number of faces per cell of sucha froth is 10, since nN < 6. It is known that the minimal number of neighboursper cell is 8 for an Euclidean froth. Thus an Euclidean froth with 8 nN < 10necessarily contains local topological defects of the kind shown in Fig.(11).Recall that the shell-network is the superposition of two elliptic 2D froths, the in-coming and the outgoing froths. The pattern of edges constituting the shell-networksets the value of nN. Therefore, Eq.(4.1) allows us to construct systematically 3DEuclidean shell- structured-inflatable froths starting from the 2D shell-networks.

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The simplest 2D froth is the hexagonal lattice. The examples displayed in Figs.(4),(5) and (6) illustrate the construction of ordered, monotiled 3D froths from a shell-network generated by different superpositions (cf. Figs.(4a), (5a) and (6a)) of twohexagonal lattices. Fig.(4b, c) shows two 3D unit cells constructed from the network(4a) (see also [12]). The cell (4b) is topologically equivalent to Kelvins tetrakaidec-

ahedron [7] [8](it builds up the Kelvin froth shown in Fig.(3)), and the cell (4c), to itstwisted variant [13]. Both structures have f = 14 and nN = 5, Eq.(3.15) givess = 2. They are indeed Euclidean space-fillers.

Fig.(5a) shows part of a shell-network with 5-sided faces, generated by the super-position of two squeezed hexagonal lattices (see also [12]). Figs.(5b) and (5c) showtwo 3D unit cells constructed from the network (5a). These cells have again f = 14.The unit cell (5b) is topologically equivalent to the tetrakaidecahedron (the Williamscell [14]). It has nN = 5, and is an Euclidean space-filler according to Eq.(3.15).

The unit cell of Fig.(5c) is topologically equivalent to the 14-sided cell (the Gold-berg cell [15]) which occurs, among others, in clathrates [8], in t.c.p. structures [16][17] and in the minimal froth of Weaire and Phelan [18]. Space can be filled layer by

layers of Goldberg cells only. The layers (Fig.(5) ) are Euclidean and the network (5a)is the same as that of the Williams space-filler. However, successive layers are moreand more distorted [19], as shown in Fig.(5d). This distortion, which stretches thenetwork in one direction and compresses it in the other, strongly suggests that we arefilling hyperbolic 3D space with a stack of Euclidean layers. It is possible to prove thiscontention by filling space shell by shell instead of layer by layer. When doing so, onefinds that most of the shell-network is composed of pentagons (12 out of 14 in each 3Dcell), but a finite density of hexagons (2 out of 14 in each 3D cell) is needed in orderto close a shell. Thus, nN > 5 which, according to Eq.(3.15), implies s > 2. Hence,the 3D manifold tiled by Goldberg cells is hyperbolic.

With another intersection of the two squeezed hexagonal lattices, one generatesthe shell-network shown in Fig.(6a). The corresponding 3D unit cell (6b), has f = 16(8 quadrilaterals, 6 hexagons and 2 octagons) and nN = 4.8. As far as we know, thisunit cell is a new monotile Euclidean space-filler.

Fig.(7) shows an example of an Euclidean shell-structured-inflatable froth madeof two different cells [8]. The shell-network (7a) has also two different tiles. Theassociated 3D unit cell (7b) has f = 12.

One can show in general that any Euclidean shell-structured-inflatable froth madewith topologically identical cells can be constructed from a shell-network generated bysuperposition of two hexagonal lattices.

The construction of 3D disordered froths from 2D disordered shell-networks is

discussed in Appendix C.Although a construction of 3D froths layer by layer has been given in [12], it mustbe emphasized that our approach, combining spherical shells with the logistic map,is more general and provides a unifying way to deal with 3D space-filling structures,whether regular or not, whatever the curvature of the manifold which they are tiling.

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V. BOUNDS ON TOPOLOGICAL PROPERTIES OF NATURAL

CELLULAR SYSTEMS AND T.C.P. STRUCTURES

The average number (n) of edges per face of a 3D froth is in general differentfrom the average number of edges per face in the shell-network ( nN). For example,the froths in Figs.(4) and (5b) have nN = 5 and n = 5.14, the froth in Fig.(6) hasnN = 4.8 and n = 5.25 and the froth in Fig.(7) has nN = 5.33 and n = 5.

The value of n is related to the average number of faces per 3D cell by

f = 126 n . (5.1)

It is interesting to study the competition between Eq.(5.1) and the Euclidean space-filling condition given by Eq.(4.1). These two relations f(nN) (labelled space-filling) and f(n) (labelled 3D cell) are plotted in Fig.(8). They meet at thepoint (n, f) given by

n = 10 + 2

73

f = 8 + 2

7

. (5.2)

It is only when the equality n = nN = n (which corresponds to f = 13.29...)is satisfied that an arbitrary cell has the freedom to adhere to a preexisting shell by anysubset of it faces, without adjustment. This freedom grants therefore a larger number ofpossibilities for building up a froth and it maximizes the orientational entropy per cell.Indeed, Eq.(5.1) is a constraint on any single 3D cell, whereas Eq.(4.1) is a constrainton the set of 3D cells in a layer. When n = nN = n, one of the two constraintsis automatically satisfied by the other and the orientational entropy is increased [20].

Note that the value f

= 13.29... falls within the range of several already knownbounds. It is consistent with the values 13.2 and 13.33... resulting from the decurvingof the dodecahedral packing with 14- and 18-sided cells or 14- and 16-sided cells,respectively [21]. Kusner [22] has shown that a single cell with minimal interfacesin a froth which is locally Euclidean or hyperbolic cannot have less than 13.39... faceson average. It is also known that the minimal number of faces per cell of a periodic,monotiled froth is 14. Weaire and Phelan have recently given an example of froth withf = 13.5 (the so-called A15 phase) which minimizes the total interfacial area [18](seealso [23]).

Natural froths minimize their free energy (Configurational Energyminus Temper-ature Entropy). With the bounds given above, this condition is realized when thevalue off is between 13.29... and 13.5 (or 14 for periodic monotiled froths). The lowerbound corresponds to configurations with maximal orientational entropy, whereas theupper bound corresponds to configurations with minimal interfacial energy.

There exists a class of natural structures, the Frank and Kasper phases (or thelarger class of t.c.p. structures [16], [17]), for which f falls within these bounds.

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These structures are periodic and made of 12-, 14-, 15- and 16-sided cells whose facesare either pentagons or hexagons. It can be verified that some of them fulfil thecondition of Euclidean space-filling given by Eq.(4.1). We can therefore assume thatthe t.c.p. structures are Euclidean shell-structured- inflatable froths. Then, their shell-network is a periodic tiling made of pentagons and hexagons only. Let the 2D unit

cell of the shell-network consist of f(5)

pentagons and f(6)

hexagons, belonging to N

polyhedra within the layer between two subsequent shells. The number of polyhedrain the 3D unit cell is a multiple of N. The average number of edges per face in theshell-network is

nN = 6f(6) + 5f(5)

f(6) + f(5). (5.3)

Substituting into Eq.(4.1), one obtains

f = 20f(6) + 14f(5)

2f(6) + f(5). (5.4)

The number of polyhedra in the 3D unit cell can be calculated with the help of thenumbers of faces the outgoing (f+) and incoming (f) froths in the unit cell ofthe shell-network. These numbers coincide with the numbers of polyhedra in the layersabove (f+) and below (f) the shell which have one or more faces belonging to the 2Dunit cell. In the limit of large shell-networks, one has the relation v+() 2f+(), withv+ (resp. v) counting the number of 3-connected vertices in the 2D unit cell whichbelong to the outgoing (resp. incoming) froths. Eq.(3.10) can then be written interms of the quantities associated with the 2D unit cell only

f+ + f = vnN 4

6 nN(5.5)

(v counts the number of 4-connected vertices in the 2D unit cell). On the other hand,since the 3D system is periodic, one has f+ +f = 2N

on average. Therefore Eq.(5.5)can be written as

N =v2

1 + 2

f(6)

f(5)

. (5.6)

If one puts into Eq.(5.4) the simplest combinations of integers f(5) and f(6) which aresuch that f falls within the two bounds 13.29... and 13.5, one retrieves the averagenumber of faces of the 3D unit cell of all experimentally known t.c.p., which are listed

in Table 1. (The Table gives all the possible combinations (f(5)

, f(6)

) up to f(6)

= 4and, for f(6) 4, only those corresponding to known natural structures.) Also givenare the corresponding values of N, obtained from Eq.(5.6). These values of N areexactly equal to the sum of the lowest non congruent numbers of 16- (p), 15- (q), 14-(r) and 12-sided polyhedra (x) in the structural formula of the corresponding t.c.p.

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[17]. The Table presents also several simple combinations (f(6), f(5)) which correspondto structures not (yet) observed (they are indicated by blanks in the last column).Notably, combinations (2,23), (2,25)... may be good candidates for t.c.p. structuresyet to be observed. On the other hand, combinations (2,19), (3,28), (3,29) and (4,39)may be too distorted to qualify as t.c.p. structures. They may be realized with atoms of

very different sizes. Note finally, that when f is represented as a function of the ratiof(5)/f(6), the structures in the Table 1 tend to gather into distinct groups. This mayindicate either the existence of unfavourable configurations or structural mode-lockinginto the simplest t.c.p. structures (A15, Z, , ... C15).

All these facts strongly suggests that the t.c.p. are shell-structured-inflatablefroths.

VI. CONCLUSION

In this paper we have introduced a new way to study froths, which emphasizestheir shell structure. We have studied an important subclass of shellstructured froths,

i.e. those which can be generated in a recursive way according to an inflationary pro-cedure. For 2D froths (and networks with any coordination number) and 3D frothswe have found that this recursive procedure is described by the logistic map. Thismap allows for a natural differentiation between froths tiling elliptic, hyperbolic orEuclidean manifolds, without any apriori imposed curvature condition. In particular,the logistic map is able to characterize 3D curved manifolds, thereby providing a wayto define the curvature from topological considerations when the GaussBonnet theo-rem is not applicable. The logistic map in 3D enables us recover known space-fillingconfigurations, and also to suggest new ones. It is clear that the approach using thelogistic map is very powerful, since classification of 3D spacefilling configurations is

reduced to the study of the 2D tiling of the (elliptic) shellsurface. As an example ofthe power and generality of this approach, we have been able to retrieve the topologicalproperties of all experimentally-known t.c.p. structures by studying the tiling of theshellsurface by pentagons and hexagons.

ACKNOWLEDGEMENTS

We are grateful to D. Weaire for many discussions, and to the referee for making usstrengthen our conclusion. This work has been supported in part by the E.U. HumanCapital and Mobility Program Physics of Foams, ref. CHRX-CT-940542.

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APPENDICES

A INFLATION OF TWO DIMENSIONAL Z-VALENT NETWORKS

WITH Z 4The generalization of Eq.(2.3) to the description of 2D shell-structured-inflatable

froths with coordination number z 4 is as follows. Every shell has (z 1) differenttypes of vertices. Extending the notation of Section 2, the various types of vertices arelabelled by V

(t)a with a = 0, 1,...,z 2, V(t)a being the number of vertices belonging to

shell (t) from which a (resp. z 2 a) edges are pointing towards shell (t + 1) (resp.shell (t 1)). Every vertex V(t)a adds a cells between shells (t) and (t + 1). The totalnumber of cells F(t) between the two shells is

F(t) =z2a=0

aV(t)a =z2a=0

(z a 2)V(t+1)a . (A.1)

Let

n

denote the average number of edges per cell layer (t). If one sums over all cells

in this layer, one obtains

nF(t) =z2a=0

(a + 1)V(t)a +z2a=0

(z a 1)V(t+1)a . (A.2)

Since

a + 1 =

1 +1

z 2

a + 1

z 2

(z a 2) (A.3)and

z a 1 = 1 +1

z

2(z a 2) +

1

z

2a , (A.4)

one has

nF(t) =

1 +1

z 2 z2a=0

aV(t)a + 1

z 2 z2a=0

(z a 2)V(t)a

+ 1

z 2 z2a=0

aV(t+1)a +

1 +1

z 2 z2a=0

(z a 2)V(t+1)a

=

1 +1

z 2

F(t) + 1

z 2

F(t1)

+ 1z 2F

(t+1) + 1 + 1z 2F

(t) .

(A.5)

One obtains finally the recursion relationn(z 2) 2(z 1)

F(t) = F(t+1) + F(t1) . (A.6)

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The matrix form of this recursion relation is the same as for z = 3 (Eq.(2.3) ), withrecursion parameter s = n(z2)2(z1). The initial conditions are F(1) = (z2)nand F(0) = 1.

Euclidean tilings are associated with the fixed point s = 2, i.e. to the equation

n =2z

z 2 . (A.7)

The only regular solutions (z and n integers) of this equation are (6, 3) (tiling bytriangles), (4, 4) (tiling by squares) and (3, 6) (tiling by hexagons) (dual of (6, 3)).

B.1 SHELL-STRUCTURED BUT NON-INFLATABLE 2D FROTHS

Some 2D shell-structured froths cannot be constructed according to the recursionprocedure of Eq.(2.3). These froths have local inclusions which are topological defectsin the recursion procedure. An inclusion in a layer is a cell with neighbouring cells inthis layer and only in one of the two neighbouring layers. Topological defects fall in

two classes: vertex decorations (Figs.(9a) and (9b)) and edge decorations (Fig.(9c)).In all cases the inclusion is on the + side of shell (t).

Defects can be eliminated by removing one or more edges and its surroundingvertices. A vertex-decoration defect is then replaced by an ordinary vertex (Fig.(10a)).An edge-decoration defect is then replaced by edges on the shell (Fig.(10b)).

The removal of one edge reduces by one unit the number of faces in the layer.This operation corresponds to the transformations E E3, V V 2, F F1.Consequently, since n = 2E

F, the average number of edges per cell changes as

n = n + 1F

1

(n 6) . (B.1.1)

The recursion parameter s = n 4 changes therefore as

s = s +1

F 1 (s 2) . (B.1.2)

One sees that the fixed point s = 2 remains unchanged by defect elimination. More-over, elliptic froths become more elliptic (i.e. n < n < 6) whereas hyperbolicfroths become more hyperbolic (i.e. n > n > 6). Thus, the Euclidean, hyperbolicor elliptic character of the manifold tiled by the froth is not modified by defect elim-ination (it is indeed given by the Euler-Poincare characteristic which is a topologicalinvariant).

B.2 SHELL-STRUCTURED BUT NON-INFLATABLE 3D FROTHS

By analogy with the 2D case, one can define a topological distance r between twocells A and B as the minimal number of faces that must be crossed by a path that

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connects A and B. A 3D shell-structured-inflatable froth is defined by the followingtwo conditions :1) For any set of cells equidistant from a germ cell, there exists a closed non self-intersecting surface that cuts these cells and no others.2) Any cell at distance t from the germ cell is the neighbour of at least one cell at

distance t + 1.Shells are closed surfaces tiled by faces of cells; they bound layers of equidistant cells.It is possible to connect two adjacent shells (t) and (t + 1) through a set of faces, eachwith one edge on shell (t) and one on shell (t + 1). Shell (t) separates the whole frothinto an inner froth, constituted of cells at distances r t, and an outer froth, withcells at distances r > t.

There are local defects which violate rules (1) or (2). These non-inflatable con-figurations in 3D froths are shown in Fig.(11). These are particular examples of thethree general classes of 3D topological defects: vertex, edge and face decoration. as in2D these non-inflatable configuration can be eliminated. Defects elimination is madeby removing one (or more) face(s), together with surrounding edges and vertices. The

removal of one face with n edges reduces by one unit the total number C of cells.This operation corresponds to the transformation C C 1 and F F 1 n.Consequently, since f = 2F

C, the average number of faces per cell changes as

f = f + 1C 1

f 2(n + 1)

. (B.2.1)

In contrast to the 2D case, this transformation depends on the parameter n. This isnot surprising since it is well-known that in 3D the value of f is not directly relatedto the curvature of the manifold tiled by the froth.

C RANDOM 3D EUCLIDEAN FROTHS FROM

2D RANDOM SHELL NETWORKS

Eq.(4.1) implies that a 3D random froth can be constructed from the superpositionof two 2D random froths. To study this general case it is useful to rewrite Eq.(4.1) interm of the number p of intersections of edges of the incoming froths by edges of theoutgoing froths and vice versa. For a given shell t this quantity is equal to

p =2V

(t)

E(t)+ + E

(t)

=2

3

2V(t)

(V(t)+ + V

(t)

), (C.1)

where we used the identity 3V

(t)

+() = 2E

(t)

+(). Using equation (3.10) it is possible toexpress p in terms of nN. One has

p =2

3

6 nNnN 4

4nN(nN 4)(Vt+ + Vt)

. (C.2)

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When the number of network cells is much larger than unity, one has p = (2/3)(6 nN)/(nN 4). Substituting into (4.1), one obtains

f = 10 + 6p . (C.3)

In principle, in random froths, p

can take any value between zero and infinity(but only between 2/3 and 1 for periodic monotiled froths). For example, p = corresponds to a froth made with layers of infinitely long bricks disposed, layer bylayer, with orientation alternating by 90o. In this case the network is a square lattice.The opposite limit (p = 0) corresponds, for example, to a 3D froth made with layersof large and small cells, when the ratio between the cell sizes tends to infinity. In thiscase the network is the result of the superposition of a froth with cells of large sizesand a froth of small sizes and the probability of intersection of two edges of these twofroths tends to zero.

REFERENCES

[1] D. Weaire and N. Rivier, Contemp. Phys. 25 (1984) 59.[2] J. Stavans, Rep. Prog. Phys. 54 (1993) 733.[3] H. G. Schuster, Deterministic Chaos (Physik-Verlag, Weinheim 1984).[4] P. M. Oliveira, M. A. Continentino and E. V. Anda, Phys. Rev. B29 (1984) 2808.

B. K. Southern, A. A. Kumar, P. D. Loly and M-A. S. Tremblay, Phys. Rev. B27 (1983) 1405. D. A. Lavis, B. W. Southern and S. G. Davinson, J. Phys. C 18(1985) 1387.

[5] see e.g. E. Kreyszig, Differential Geometry (Dover, New York, 1991).[6] T. Aste and N. Rivier, J. Phys. A 28 (1995) 1381.[7] W. Thomson (Lord Kelvin, Phil. Mag. 24 (5) (1887) 503.[8] R. Williams, The Geometrical Foundation of Natural Structure Dover, New York,

1979).[9] B. R. Pollard, An Introduction to Algebraic Topology (University of Bristol Press

1977).[10] S. S. Chern, Ann. Math. 45 (1944) 747.[11] H.M.S. Coxeter, Regular Polytopes (Dover, New York, 1973).[12] J. A. Glazier and D. Weaire, Phil. Mag. Lett. 70 (1994) 351.[13] D. Weaire and R. Phelan, Phil. Mag. Lett. 70 (1994) 351.

[14] R. E. Williams, Science161

(1968) 276.[15] M. Goldberg, Tohoku Math. J. 40 (1934) 226.[16] F.C. Frank and J.S. Kasper, Acta crystallogr. 12 (1959) 483.[17] D.P. Shoemaker and C.B. Shoemaker, Acta crystallogr. B42 (1986) 3.[18] D. Weaire and R. Phelan, Phil. Mag. Lett. 69 (1994) 107.

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[19] D. Weaire, private communication.[20] N. Rivier and A. Lissowski, J. Phys. A 15 (1982) L143.[21] J.F. Sadoc and R. Mosseri, J. Phys. (Paris) 46 (1985) 1809.[22] R. Kusner, Proc. R. Soc. Lond. A 439 (1992) 683.[23] N. Rivier, Phil. Mag. Lett. 69 (1994) 297.

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FIGURE CAPTIONS

Fig.1

Schematic picture of a 2D shell-structured-inflatable froth.

Fig.2

Any 3D shell (t) is tiled with a network generated by the intersection of the facescoming to and going away from its surface. This shell-network has 4-connected vertices

V(t)

(whose all 4 edges are belonging to the shell-network) and 3-connected vertices

V(t)+() (with 3 edges belonging to the shell-network and the last one going away from

it).

Fig.3

An example of 3D shell-structured-inflatable froth, the Kelvin froth. A portion ofthe shell-network is brought out by hatcheries.

Fig.4

The two 3D space-filling unit cells constructed from the shell-network (a). Thecell (b) is topologically equivalent to Kelvins tetrakaidecahedron and the cell (c) toits twisted variant. Both have 14 faces.

Fig.5The two 3D space-filling unit cells constructed from the shell-network (a) gener-

ated by the superposition of two squeezed hexagonal lattices. The cell (b) is topolog-ically equivalent to the tetrakaidecahedron. The cell (c) is topologically equivalentto the Goldberg cell.

Fig.6

The 3D space-filling unit cell (b) (which has 16 faces) resulting from the shell-network (a) generated by the superposition of two squeezed hexagonal lattices.

Fig.7

Example of a 3D periodic shell-structured-inflatable froth (with f = 12) whoseunit cell has two different elementary cells. (a) Shell-network. (b) 3D unit cell.

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Fig.8

The average number f of faces per cell in a froth plotted as a function of theaverage number nN of edges per 2D cell in the shell network (Eq.(4.1), curve labelledspace-filling) and of the average number n of edges per face in the froth (Eq.(5.1),curve labelled 3D). The abscissa n represents both

nN and

n.

Fig.9

Local topological defects in the 2D recursion procedure. Figures (a) and (b) areexamples of vertex decorations whereas Figure (c) is an example of edge decoration.The index t denotes the topological distance.

Fig.10

Schematic representations of the elimination of a 2D local topological defect. (a)

Vertex decoration. (b) Edge decoration.

Fig.11

Local topological defects in the 3D recursion procedure, (a) is a vertex decorationdefect, (b) is an edge decoration defect and (c) is a face decoration defect. The indext denotes the topological distance.

TABLE CAPTION

Table 1

Average number of faces f and (minimal) number of elements in the 3D unitcell N of all the t.c.p. structures known experimentally (labelled in the last column)and of hypothetical t.c.p. structures (indicated by a blank in the last column). Theintegers p, q, r and x indicate respectively the proportions of 3D cells with 16, 15, 14and 12 faces present in the 3D unit cell.

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Number of

hexagons

Number of

pentagons N* p q r x t.c.p.

1 10 13.33333 3 1 0 0 2 C15; C14

1 11 13.38462 13 2 2 2 7 p; K7Cs6; ; M

1 12 13.42857 7 0 2 2 3 Z

2 24 13.42857 14 1 2 5 6 P; 1 13 13.46667 15 0 2 8 5 ; H

1 14 13.5 4 0 0 3 1 A15

2 19 13.30435 23

2 21 13.36 25 6 2 2 15 C

2 23 13.40741 27

2 25 13.44828 29

2 27 13.48387 31

3 28 13.29412 17

3 29 13.31429 35

3 31 13.35135 37 10 2 2 23 X

3 32 13.36842 19 4 2 2 11 I3 34 13.4 20

3 35 13.41463 41

3 37 13.44186 43

3 38 13.45455 11 0 2 5 4 J

3 40 13.47826 23

3 41 13.48936 47

4 39 13.31915 47

4 41 13.34694 49

4 43 13.37255 51

4 45 13.39623 53 8 6 12 27 R

4 47 13.41818 55 7 4 19 25 K*7 90 13.46154 52 0 4 13 9 F

9 92 13.34545 55 16 2 2 35 Mg4Zn711 142 13.46341 41 7 4 19 25 K

13 136 13.35802 81 20 6 6 49 T

13 136 13.35802 81 23 0 9 49 SM

13 160 13.44086 93 6 10 40 37

TABLE 1


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13

13.1

13.2

13.3

13.4

13.5

13.6

13.7

13.8

13.9

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t. aste, d. boose and n. rivier- from one cell to the whole froth: a whole map

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